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The previous lower bound, due to Blaser, was 5n^2/2-3n (the case p=1).\n  The new bounds improve Blaser's bound for all n>84. I also prove lower bounds for rectangular matrices significantly better than the the previous bound."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1206.1530","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2012-06-07T15:36:36Z","cross_cats_sorted":["math.AG","math.RT"],"title_canon_sha256":"18a5204d8c066bf027b7fe1fd12f829833a6bf0f600f0c1da000c5b2438bdbde","abstract_canon_sha256":"58a4f68b4b85aaa90634f746ca324ebd4eed539f79cbe3113ebc28f4ffbf5ae3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:08:27.362976Z","signature_b64":"8+SEghtZHIodjoE6tFfJ4378FLex8GXufx/0cdNXSkolf2T7A04Nq5cvtLlNMyZd/h/ZYhImef/Vbz50fRhKAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"178ffc6201b318af561a741dded00c3e2c5c7cf0d230722ee1c138ac772f39c9","last_reissued_at":"2026-05-18T03:08:27.362260Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:08:27.362260Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"New lower bounds for the rank of matrix multiplication","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.RT"],"primary_cat":"cs.CC","authors_text":"J. 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I also prove lower bounds for rectangular matrices significantly better than the the previous bound."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.1530","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1206.1530","created_at":"2026-05-18T03:08:27.362360+00:00"},{"alias_kind":"arxiv_version","alias_value":"1206.1530v2","created_at":"2026-05-18T03:08:27.362360+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.1530","created_at":"2026-05-18T03:08:27.362360+00:00"},{"alias_kind":"pith_short_12","alias_value":"C6H7YYQBWMMK","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_16","alias_value":"C6H7YYQBWMMK6VQ2","created_at":"2026-05-18T12:27:01.376967+00:00"},{"alias_kind":"pith_short_8","alias_value":"C6H7YYQB","created_at":"2026-05-18T12:27:01.376967+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/C6H7YYQBWMMK6VQ2OQO55UAMHY","json":"https://pith.science/pith/C6H7YYQBWMMK6VQ2OQO55UAMHY.json","graph_json":"https://pith.science/api/pith-number/C6H7YYQBWMMK6VQ2OQO55UAMHY/graph.json","events_json":"https://pith.science/api/pith-number/C6H7YYQBWMMK6VQ2OQO55UAMHY/events.json","paper":"https://pith.science/paper/C6H7YYQB"},"agent_actions":{"view_html":"https://pith.science/pith/C6H7YYQBWMMK6VQ2OQO55UAMHY","download_json":"https://pith.science/pith/C6H7YYQBWMMK6VQ2OQO55UAMHY.json","view_paper":"https://pith.science/paper/C6H7YYQB","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1206.1530&json=true","fetch_graph":"https://pith.science/api/pith-number/C6H7YYQBWMMK6VQ2OQO55UAMHY/graph.json","fetch_events":"https://pith.science/api/pith-number/C6H7YYQBWMMK6VQ2OQO55UAMHY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/C6H7YYQBWMMK6VQ2OQO55UAMHY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/C6H7YYQBWMMK6VQ2OQO55UAMHY/action/storage_attestation","attest_author":"https://pith.science/pith/C6H7YYQBWMMK6VQ2OQO55UAMHY/action/author_attestation","sign_citation":"https://pith.science/pith/C6H7YYQBWMMK6VQ2OQO55UAMHY/action/citation_signature","submit_replication":"https://pith.science/pith/C6H7YYQBWMMK6VQ2OQO55UAMHY/action/replication_record"}},"created_at":"2026-05-18T03:08:27.362360+00:00","updated_at":"2026-05-18T03:08:27.362360+00:00"}